6 3 GRAPHING SINE AND COSINE FUNCTIONS Periodic

6. 3 – GRAPHING SINE AND COSINE FUNCTIONS

Periodic Function and Period A function is periodic if, for some real number α, f(x + α) = f(x) for each x in the domain of f. The least positive value of α for which = f(x + α) is the period of the function. f(x)

Which function is periodic?

Determine the period of the function

Graph of Sine Look at the value for sin(x) for the domain value between -2π and 2π in multiples of π/4 x sin(x) -2π 0 0 0 -7π/4 √ 2/2 -3π/2 1 -5π/4 √ 2/2 3π/4 √ 2/2 -π 0 -3π/4 -√ 2/2 5π/4 -√ 2/2 -π/2 -1 3π/2 -1 -π/4 -√ 2/2 7π/4 -√ 2/2 2π 0

Graph of Sine

Properties of the Graph of Sine What is the domain? All real numbers What is the range? Set of real numbers between -1 and 1 What is the period? 2π Where all the x- at πn, where n is any integer intercepts? Where is the y-intercept? at (0, 0) Where are the maximums? At y = 1 and occur when x = π/2 + 2πn Where are the minimums? At y = -1 and occur when x = 3π/2 + 2πn

Find each value by referring to the graph of sine 1. sin(5π/2) 2. sin(3π)

Find the values of θ for which sin θ = -1 is true.

Graph y = sinx for the interval of 2π to 4π

Graph of Cosine Look at the value for cos(x) for the domain value between -2π and 2π in multiples of π/4 x cos(x) -2π 1 0 1 -7π/4 √ 2/2 -3π/2 0 -5π/4 -√ 2/2 3π/4 -√ 2/2 -π -1 -3π/4 -√ 2/2 5π/4 -√ 2/2 -π/2 0 3π/2 0 -π/4 √ 2/2 7π/4 √ 2/2 2π 1

Graph of Cosine

Properties of the Graph of Cosine What is the domain? All real numbers What is the range? Set of real numbers between -1 and 1 What is the period? 2π Where all the x- at π/2 + πn, where n is any integer intercepts? Where is the y-intercept? at (0, 1) Where are the maximums? At y = 1 and occur when x = πn, n is an even integer Where are the minimums? At y = -1 and occur when x = πn, n is an odd integer

Homework / In Class Work Pg 363 -366: #13 -33 odds 37, 38, 39